# Definition:Continuous Mapping (Topology)/Set

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## Definition

Let $T_1 = \left({S_1, \tau_1}\right)$ and $T_2 = \left({S_2, \tau_2}\right)$ be topological spaces.

Let $f: S_1 \to S_2$ be a mapping from $S_1$ to $S_2$.

Let $S$ be a subset of $S_1$.

The mapping $f$ is **continuous on $S$** if and only if $f$ is continuous at every point $x \in S$.

## Also see

- Results about
**continuous mappings**can be found here.